Noisy Rumor Spreading and Plurality Consensus

Emanuele Natale†

joint work withPierre Fraigniaud∗

ACM Symposium onPrinciples of Distributed Computing

July 25-29, 2016Chicago, Illinois

†∗

Rumor-Spreading Problem

FOOD!

Rumor-Spreading Problem

Plurality Consensus Problem

This way! This way!

THIS WAY!

Plurality Consensus Problem

Some examples (Plurality Consensus)Flocks of birds [Ben-Shahar et al. ’10]

Some examples (Plurality Consensus)

Schools of fish [Sumpter et al. ’08]

Flocks of birds [Ben-Shahar et al. ’10]

Some examples (Plurality Consensus)

Schools of fish [Sumpter et al. ’08]

Insects colonies [Franks et al. ’02]

Flocks of birds [Ben-Shahar et al. ’10]

Some examples (Plurality Consensus)

Schools of fish [Sumpter et al. ’08]

Eukaryotic cells [Cardelli et al. ’12]

Insects colonies [Franks et al. ’02]

Flocks of birds [Ben-Shahar et al. ’10]

Animal Communication Despite Noise

Noise affects animal communication,but animals cannot use coding theory...

Animal Communication Despite Noise

O. Feinerman, B. Haeupler and A. Korman.Breathe before speaking: efficient informationdissemination despite noisy, limited and anonymouscommunication. (PODC ’14)

Noise affects animal communication,but animals cannot use coding theory...

Natural rules efficiently solve rumor spreading andplurality consensus despite noise.

=⇒

Animal Communication Despite Noise

O. Feinerman, B. Haeupler and A. Korman.Breathe before speaking: efficient informationdissemination despite noisy, limited and anonymouscommunication. (PODC ’14)

Noise affects animal communication,but animals cannot use coding theory...

They only consider the binary-opinion case.Our contribution: generalize to many opinions.

Natural rules efficiently solve rumor spreading andplurality consensus despite noise.

=⇒

Binary Case - Model

n agents. One agent has one bit to spread.

Binary Case - Model

Communication model: PUSH model [Pittel ’87]:at each round each agent can send a bit to anotherone chosen uniformly at random.

????????

Binary Case - Model

Communication model: PUSH model [Pittel ’87]:at each round each agent can send a bit to anotherone chosen uniformly at random.

Binary Case - Model

Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).

!

Binary Case - Model

Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).

Binary Case - Model

Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).

Binary Case - Model

Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).

!

Breathe Before Speaking

blue vs red:1/0

trivialstrategy

Breathe Before Speaking

blue vs red:2/0

trivialstrategy

Breathe Before Speaking

blue vs red:3/1

trivialstrategy

Breathe Before Speaking

blue vs red:9/6 = 1.5

trivialstrategy

Breathe Before Speaking

blue vs red:18/13 ≈ 1.4

trivialstrategy

Breathe Before Speaking

blue vs red:35/29 ≈ 1.2

trivialstrategy

Breathe Before Speaking

blue vs red:35/29 ≈ 1.2

trivialstrategy

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:1/0

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:1/0

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:1/0

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:1/0

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:3/1

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:3/1

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:3/1

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:8/4

Breathe Before Speaking

“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”

(Razin et al. ’13)

Stage 1: Spreading

blue vs red:40/24 ≈ 1.7

Breathe Before Speaking

Stage 1: Spreading

blue vs red:40/24 ≈ 1.7

Stage 2: Amplifying majority

# > # ?# < # ?

Mathematical Challenges

• Stochastic Dependence

Mathematical Challenges

• Stochastic Dependence

Mathematical Challenges

• Stochastic Dependence

• “Small Deviations”

Pr(X ≥ t) ≤ f(t)

X

Pr(X)

0

Pr(X ≥ t) ≥ 12 + g(t)

g(t), f(t) << 1

Multivalued Case

!

Multivalued Case

Multivalued Case

Multivalued Case

!

Multivalued Case

Noise Matrix:

∼P :=

p ,

p ,

p ,

p , p ,

p ,

p , p ,

p ,

Multivalued Case

δ-majority-biased configuration w.r.t. :

Noise Matrix:

Configuration c := (# /n,# /n,# /n)

∼P :=

p ,

p ,

p ,

p , p ,

p ,

p , p ,

p ,

# /n−# /n > δ

# /n−# /n > δ

Majority-Preserving Matrix

Randomsenderin conf. c

Noise actingaccording tomatrix P

Messagedistributedas c · P

Majority-Preserving Matrix

(cP ) − (cP ) > εδ

(cP ) − (cP ) > εδ

(ε, δ)-majority-preserving noise matrix:

(cP ) − (cP ) > εδ

(cP ) − (cP ) > εδ

Randomsenderin conf. c

Noise actingaccording tomatrix P

Messagedistributedas c · P

Main Result

Theorem. Let S be the initial set of agents withopinions in [k]. Suppose that S is δ =Ω(√

logn/|S|)-majority-biased with |S| = Ω( lognε2 )

and the noise matrix P is (ε, δ)-majority-preserving.Then the plurality consensus problem can be solvedin O( logn

ε2 ) rounds w.h.p., with O(log logn+ log 1ε )

memory per node.

Main Result

Theorem. Let S be the initial set of agents withopinions in [k]. Suppose that S is δ =Ω(√

logn/|S|)-majority-biased with |S| = Ω( lognε2 )

and the noise matrix P is (ε, δ)-majority-preserving.Then the plurality consensus problem can be solvedin O( logn

ε2 ) rounds w.h.p., with O(log logn+ log 1ε )

memory per node.

|S| = 1 =⇒ rumor spreading in O( lognε2 ) rounds

Main Result

Theorem. Let S be the initial set of agents withopinions in [k]. Suppose that S is δ =Ω(√

logn/|S|)-majority-biased with |S| = Ω( lognε2 )

and the noise matrix P is (ε, δ)-majority-preserving.Then the plurality consensus problem can be solvedin O( logn

ε2 ) rounds w.h.p., with O(log logn+ log 1ε )

memory per node.

P =(

1/2 + ε 1/2− ε1/2− ε 1/2 + ε

)=⇒ Feinerman et al.

|S| = 1 =⇒ rumor spreading in O( lognε2 ) rounds

Poisson Approximation

?∼ c · P

Poisson Approximation

?∼ c · P

? ∼Poisso

n((c ·P )i)

Poisson ApproximationLemma. balls-in-bins experiment:- h colored balls are thrown in n bins, hi balls have color1 ≤ i ≤ k,- Xu,iu∈1,...,n,i∈1,...,k number of i-colored balls that end upin bin u,- f non-negative function with Z≥0 argumentsxu,iu∈1,...,n,i∈1,...,k and z,- Yu,iu∈1,...,n,i∈1,...,k independent r.v. withYu,i ∼Poisson(hi/n) and Z integer valued r.v. independent fromXu,is and Yu,is.

E [f (X1,1, ..., Xn,1, Xn,2, ..., Xn,k, Z)]

≤ ek√∏

i hi E [f (Y1,1, ..., Yn,1, Yn,2, ..., Yn,k, Z)] .

Corollary. Given conf. c, if event E holds in process P withprob 1− n−b with b > (k log h)/(2 logn), then it holds w.h.p.also in the original process.

Probability Amplification

A dice with k faces is thrown ` times.

prob

abili

ty

γ

1 2 . . . face

Probability Amplification

A dice with k faces is thrown ` times.

M := most frequent face in the ` throws(breaking ties at random).

For any j 6= 1

Pr (M = 1)− Pr (M = j) ≥ const ·√` γ(1− γ2)

`−12

prob

abili

ty

γ

1 2 . . . face

Probability Amplification

A dice with k faces is thrown ` times.

M := most frequent face in the ` throws(breaking ties at random).

For any j 6= 1

Pr (M = 1)− Pr (M = j) ≥ const ·√` γ(1− γ2)

`−12

prob

abili

ty

γ

1 2 . . . face

open problem: const ≈ e−Θ(k)

Binomial vs Beta

Given p ∈ (0, 1) and 0 ≤ j ≤ ` it holds

Pr (Bin(n, p) ≤ j) =∑j<i≤`

(`

i

)pi (1− p)`−i

=(

`

j + 1

)(j + 1)

∫ p

0zj (1− z)`−j−1

dz

= Pr (Beta(n− k, k + 1) < 1− p).

Binomial vs Beta

Given p ∈ (0, 1) and 0 ≤ j ≤ ` it holds

Pr (Bin(n, p) ≤ j) =∑j<i≤`

(`

i

)pi (1− p)`−i

=(

`

j + 1

)(j + 1)

∫ p

0zj (1− z)`−j−1

dz

= Pr (Beta(n− k, k + 1) < 1− p).

Multinomial vs Dirichlet?